The main difference between TaqMan and SYBR green assay is specificity.
In SYBR green assay, the dye binds to any double-stranded DNA sequence. This means that you could also detect fluorescence emitted from non-specific qPCR products, such as primer dimers. This can result in false positive results.
The TaqMan assay only measures amplification progression of the target sequence, as the probes are target specific. Also, you can monitor the quantity of various qPCR products in a single reaction by adding different primers and TaqMan probes with different reporter dyes to the master mix. This multiplex approach allows you to detect several fluorescent signals at the end of every thermal cycle.
I would recommend the use of TaqMan assay for your project as it is highly sensitive and will help in specific quantification compared to SYBR green assay. If you have enough funds, you should go for the TaqMan assay because TaqMan probes are expensive.
It is correct that the signal detection by TaqMan probes is sequence-specific, whereas dsDNA-bindingy dyes like SYBR Green detecs any dsDNA that is being amplified. However, specific detection ist not a guarantuee of specific amplification, and specific amplification is a prerequisite of correct quantitation. SYBR Green allows you to check for the amplification of wrong products in the closed PCR tube by melting curve analyses. If you use TaqMan probes, you will have to run a gel post-PCR to confirm that the amplification was specific.
The sensitivity is not any problem, neither for TaqMan nor for SYBR Green. Both methods will detect a single molecule (profided the assay is well optimized).
NB: TaqMan probes are short oligonucleotides that are labelled with a fluorescent dye (a reporter) and a quencher (usually a BHQ) a few nucleotides apart. Further, the 3` end is modified to prevent an elongation by the polymerase. Due to their close proximity, the quencher quenches the fluorescence of the reporter dye (via a resonance energy transfer from the excited delocalized electron system of the reporter to the electron system of the quencher that translates much of this energy into heat or emits it as photons with much longer wavelength than the reporter would). The probes anneal at the same time when the primers anneal, but they won't be extended. The polymerase starts extending the primers and reaches the probe that blocks the way. The Taq DNA-polymerase has a exonuclease activity that chops down such blocking oligos that are in its way. By destructing the probe, the reporter is released from the quencher and its fluorescence is not longer quenched. This chopping down the nucleotides on its way reminds a bit of PacMan eaing the pills on its way. This might have been the inspiration behind the name "TaqMan".
This assay was owned by Applied Biosystems who tried to sell this chemistry together with their instruments. There are numerous other ways to generate sequence-specific signals, like scorpion primers, hybridization probes, molecular beacons, and others.
Also note that TaqMan probes add the fluorescence intensity of one molecule per amplified molecule, whereas several SYBR Green moleculed bind to each dsDNA molecule, adding up their fluorescence intensity. Further, the free reporters from the TaqMan probes accumulate, so that the added fluorescence is added on top of a growing background after each cycle, was may negatively impact the signal-to-noise ratio.